Is the integrator of a Riemann--Stieltjes integral necessarily of bounded variation?

Question

According to https://en.wikipedia.org/wiki/Riemann%E2%80%93Stieltjes_integral, if $g$ is a real-valued function of bounded variation and $f$ is another real-valued function, then one can define the Riemann--Stieltjes integral $\int_s^t f d g = \lim_{\lvert \pi \rvert \to 0} \sum_{x,y\in \pi} f(x)(g(y)-g(x))$, where the limit is over all sequences of partitions $\pi$ of $[s,t]$ with vanishing mesh $\lvert \pi \rvert = \sup_{x,y \in \pi} \lvert v - u\rvert$. Is the converse true? I.e., if the Riemann--Stieltjes integral exists, must $g$ be of bounded variation?